Question

The wave function for a standing wave on a string is described by y(x, t) = 0.021 sin(4x) cos (56át), where y and x are in meters and t is in seconds. Determine the maximum displacement and maximum speed of a point on the string at the following positions. (a) x = 0.10 m Ymax = m Vmax = m/s (b) x = 0.25 m Ymax = Vmax = m m/s (c) x = 0.30 m Ymax = Vmax = m m/s (d) x = 0.50 m Ymax = Vmax = m m/s eBook

Answer 1

1) So y is maximum is at when cos(56pi t) =1

For that t =0, 2, 4 ...

So

$y_{max}= 0.021 sin(4\pi x)$​​​​​​

And velocity is

$v=\frac{dy}{dt}= 0.021 (56\pi)sin(4\pi x)sin(56\pi t)$

So v is maximum when t = 1/112 sec we get

$v_{max}= 3.69 sin(4\pi x)$

So at x = 0.1 m

We get,

утат 0.021sin(0.4л) = 0.02m

And

maz 3.69sin (47(0.1)) = 3.5m/s

At x = 0.25 m

$y_{max}= 0.021 sin(4\pi (0.25))=0m$

$v_{max}= 3.69 sin(4\pi (0.25))= 0 m/s$

At x = 0.30 m

$y_{max}= 0.021 sin(4\pi (0.30))=-0.0123 m$

$v_{max}= 3.69 sin(4\pi (0.30))= -2.17 m/s$

At x = 0.5 m

$y_{max}= 0.021sin(4\pi (0.50))= 0 m/s$

$v_{max}= 3.69 sin(4\pi (0.50))= 0 m/s$